An sl(n) stable homotopy type for matched diagrams.

Abstract

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sln complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer n ≥ 2, we associate a stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case n ≥ 3 the cohomology of the stable homotopy type agrees with the sln Khovanov-Rozansky cohomology of the underlying knot. We make some consistency checks of this sln stable homotopy type and show that it exhibits interesting behaviour. For example we find a CP2 in the sl3 type for some diagram, and show that the sl4 type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces.